The description provided outlines the content of a standard introductory calculus course. Here are the main problems and topics it likely includes, grouped by category:

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1. Limits of a Function

• Evaluating Limits: $\lim_{x \to a} f(x)$ Example: $\lim_{x \to 2} \frac{x^2 - 4}{x-2}$
• One-Sided Limits: $\lim_{x \to a^+} f(x), \quad \lim_{x \to a^-} f(x)$
• Limits at Infinity: $\lim_{x \to \infty} \frac{1}{x^2}, \quad \lim_{x \to -\infty} f(x)$
• Indeterminate Forms and L'Hôpital's Rule: Example: $\lim_{x \to 0} \frac{\sin x}{x}$
• Asymptotic Behavior: Vertical, horizontal, and slant asymptotes.

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2. The Derivative

• Definition of the Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Derivative Rules:
  • Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$
  • Product Rule: $(uv)' = u'v + uv'$
  • Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
  • Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
• Derivatives of Common Functions:
  • $\sin x, \cos x, \tan x, e^x, \ln x$

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3. Applications of the Derivative

• Approximations of Error Using Differentials: Example: Estimating $\sqrt{4.1}$ using $f(x) = \sqrt{x}$.
• Optimization Problems: Maximizing or minimizing functions. Example: Finding dimensions of a rectangle of maximum area.
• Curve Sketching:
  • Finding critical points.
  • Determining intervals of increase/decrease.
  • Identifying points of inflection.
  • Analyzing concavity.
• Related Rates: Example: A balloon inflates such that its radius increases at a rate of 2 cm/s. How fast is the volume increasing?
• Newton’s Method for Root Approximation: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

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4. The Definite Integral

• Definition: $\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x$
• Fundamental Theorem of Calculus:
  • Part 1: $F'(x) = f(x)$
  • Part 2: $\int_a^b f(x) \, dx = F(b) - F(a)$
• Integration Techniques:
  • Substitution.
  • Basic integration formulas: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$.
• Applications of Integration:
  • Area under a curve.
  • Finding the total distance traveled.

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5. Basic Applications of Integrals

• Area Between Curves: $\int_a^b [f(x) - g(x)] \, dx$
• Volume of Solids of Revolution: Using disk/washer method: $V = \pi \int_a^b [R(x)^2 - r(x)^2] \, dx$
• Average Value of a Function: $f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx$

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Would you like to explore the solutions or detailed breakdowns for any of these topics? Below are a few related questions for deeper understanding:

1. Can you calculate limits using algebraic simplifications or L’Hôpital’s Rule?
2. How are derivatives applied to real-world problems like optimization?
3. What’s the importance of the Fundamental Theorem of Calculus?
4. How do you sketch curves using critical points and concavity?
5. Can you solve a problem involving area under a curve using definite integrals?

Tip: Start with understanding limits, as they are foundational to all other calculus concepts.